Yes, as long as he qualifies for the loan, he can be on it and on Title. He can do a Quit Claim to take himself off Title but he will still be on the loan. The only way he can remove himself from the loan is to either pay off the loan or refinance it into your name only.
Meantime, your son can absolutely co-sign, if necessary, but a quit claim will not relieve him from liability for repaying the loan.
There are tax implications involved in this type of transaction, as mentioned elsewhere.
However, this usually is only an issue if there is a significant difference in market value between the current transfer and the original transfer.
The ability of your son to co-sign will also depend on his credit score and financial strength to to qualify for the loan. 50% sounds like a substantial downpayment. Vesting of the title will also depend upon his marital status as you should consider community property laws. if he is married, then likely, his spouse would also need to quit claim. I hope this helps.
Trulia is famous for agents and lenders answering questions with little to no information to base their answers on or assuming the wrong scenarios.
Call me of you want to go over these details that will impact how you structure this home purchase.